Picking a phone to game on 4G LTE

Hiya guys, I am in need of some advice. Tomorrow I am going to a Verizon dealer and picking up a phone. I get and average of 10mbs-15mbs download and 1-3mbs upload with my families Mifi 4G LTE in the house. They are dropping the 5GB plan and I am picking up an unlimited data plan for internet gaming and general use. I was suggested to pick up the Galaxy S3 as it can be had for just $50 and it can be used as a hotspot for my laptop/pc xbox etc.

Is there a phone out there that I should get instead through that works better than the S3 as a hotspot?

Before you start saying I should get a land line connection like Cable, DSL etc., I can't, its just not possible, otherwise I would.

Honest question here- the bandwidth may be good but how is the latency?